1. What is Complex Number Angle?

The complex number angle (also known as argument or phase) represents the direction of a complex number in the complex plane. It is measured from the positive real axis to the line connecting the origin to the complex number point.

2. How Does the Calculator Work?

The calculator uses the arctangent function:

Where:

• \( \theta \) — Angle in radians
• \( \text{Im} \) — Imaginary part of complex number
• \( \text{Re} \) — Real part of complex number

Explanation: The calculator uses atan2 function which automatically handles all quadrants and special cases, providing the correct angle between -π and π radians.

3. Importance of Complex Number Angle

Details: The angle is crucial for understanding complex numbers in polar form, signal processing, electrical engineering, and quantum mechanics. It helps in analyzing phase relationships and rotational transformations.

4. Using the Calculator

Tips: Enter both real and imaginary parts as decimal numbers. The calculator will compute the angle in both radians and degrees. At least one of the values must be non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of the angle?
A: The angle ranges from -π to π radians (-180° to 180°) using the principal value.

Q2: What happens when both inputs are zero?
A: The angle is undefined at the origin (0,0) since there is no unique direction.

Q3: How is the quadrant determined?
A: The atan2 function automatically determines the correct quadrant based on the signs of both real and imaginary parts.

Q4: What is the difference between atan and atan2?
A: atan only gives results between -π/2 and π/2, while atan2 gives the full range -π to π by considering both coordinates.

Q5: Where is complex number angle used in real applications?
A: Used in electrical engineering for AC circuit analysis, signal processing for phase analysis, control systems, and quantum mechanics for wave functions.